Combinatorics (math.CO)

Abstract: The transmission ${\rm Tr}_G(u)$ of a vertex $u$ of a connected graph $G$ is the sum of distances from $u$ to all other vertices. $G$ is a stepwise transmission irregular (STI) graph if $|{\rm Tr}_G(u) - {\rm Tr}_G(v)|= 1$ holds for any edge $uv\in E(G)$. In this paper, generalized STI graphs are introduced as the graphs $G$ such that for some $k\ge 1$ we have $|{\rm Tr}_G(u) - {\rm Tr}_G(v)|= k$ for any edge $uv$ of $G$. It is proved that generalized STI graphs are bipartite and that as soon as the minimum degree is at least $2$, they are 2-edge connected. Among the trees, the only generalized STI graphs are stars. The diameter of STI graphs is bounded and extremal cases discussed. The Cartesian product operation is used to obtain highly connected generalized STI graphs. Several families of generalized STI graphs are constructed.

Abstract: We study the Maker-Maker version of the domination game introduced in 2018 by Duch\^ene et al. Given a graph, two players alternately claim vertices. The first player to claim a dominating set of the graph wins. As the Maker-Breaker version, this game is PSPACE-complete on split and bipartite graphs. Our main result is a linear time algorithm to solve this game in forests. We also give a characterization of the cycles where the first player has a winning strategy.

Abstract: Let $G$ be a graph in which each edge is assigned one of the colours $1, 2, \ldots, m$, and let $\Gamma$ be a subgroup of $S_m$. The operation of switching at a vertex $x$ of $G$ with respect to an element $\pi$ of $\Gamma$ permutes the colours of the edges incident with $x$ according to $\pi$. We investigate the complexity of whether there exists a sequence of switches that transforms a given $m$-edge coloured graph $G$ so that it has a colour-preserving homomorphism to a fixed $m$-edge coloured graph $H$ and give a dichotomy theorem in the case that $\Gamma$ acts transitively.

Abstract: We study a version of the lights out game played on directed graphs. For a digraph $D$, we begin with a labeling of $V(D)$ with elements of $\mathbb{Z}_k$ for $k \ge 2$. When a vertex $v$ is toggled, the labels of $v$ and any vertex that $v$ dominates are increased by 1 mod $k$. The game is won when each vertex has label 0. We say that $D$ is $k$-Always Winnable (also written $k$-AW) if the game can be won for every initial labeling with elements of $\mathbb{Z}_k$. We prove that all acyclic digraphs are $k$-AW for all $k$, and we reduce the problem of determining whether a graph is $k$-AW to the case of strongly connected digraphs. We then determine winnability for tournaments with a minimum feedback arc set that arc-induces a directed path or directed star digraph.